Basic Notions of Algebra
§22. K-theory 230 A. Topological X-theory 230 Vector bundles and the functor Vec(X). Periodicity and the functors KJX). K(X) and t the infinite-dimensional linear group. The symbol of an elliptic differential operator. The index theorem. B. Algebraic K-theory 234 The group of classes of projective modules. K , K and K of a ring. K of a field and o l n 2 its relations with the Brauer group. K-theory and arithmetic. Comments on the Literature 239 References 244 Index of Names 249 Subject Index 251 Preface This book aims to present a general survey of algebra, of its basic notions and main branches. Now what language should we choose for this? In reply to the question 'What does mathematics study?', it is hardly acceptable to answer 'structures' or 'sets with specified relations'; for among the myriad conceivable structures or sets with specified relations, only a very small discrete subset is of real interest to mathematicians, and the whole point of the question is to understand the special value of this infinitesimal fraction dotted among the amorphous masses. In the same way, the meaning of a mathematical notion is by no means confined to its formal definition; in fact, it may be rather better expressed by a (generally fairly small) sample of the basic examples, which serve the mathematician as the motivation and the substantive definition, and at the same time as the real meaning of the notion.
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10 Semisimple Modules and Rings
ll Division Algebras of Finite Rank
12 The Notion of a Group
Inﬁnite Discrete Groups
Lie Groups and Algebraic Groups
16 General Results of Group Theory
17 Group Representations
Characters of compact Abelian groups and Fourier series Weyl and Ricci tensors in
19 Lie Algebras and Nonassociative Algebra
21 Homological Algebra
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Abelian groups algebraic groups analogue arbitrary automorphism axioms called classiﬁcation coefﬁcients cohomology commutative ring compact complex analytic complex numbers consider consists construction contained coordinate coordinatisation corresponding coset curve deﬁned deﬁnition denoted dimension direct sum division algebra elements equation exact sequence Example exists extension L/K ﬁnd ﬁnite ﬁnite extension ﬁnite groups ﬁnite number ﬁnite rank ﬁnite type ﬁnite-dimensional ﬁrst ﬁxed follows functor g e G Galois geometry given GL(n group G Hence homomorphism identity inﬁnite integral invariant irreducible representations isomorphic kernel lattice Lie algebra Lie groups linear transformations manifold matrix module morphisms multiplication n-dimensional normal subgroup notion number ﬁeld permutations plane properties quaternions quotient rational functions real number reﬂection relations satisﬁes semisimple sheaf simple SL(n solvable submodule subspace symmetry group tensor Theorem theory topological space unique vector ﬁelds vector space